A lot depends on your street data. If you are in the US you can use TIGER. The TIGER chain (line/street segement) include end point coordinates. If you have some DB skills you can join the chains file with itself based on the endpoints. From the resulting record you can extract the intersections.

You can do all this in Access, no need for a GIS program. Download the TIGER technical reference for details on the ASCII record layou of the TIGER records. For 99% of the cases all you need is in the RT1 file.

Do a search for "TIGER home" to find the downloadable files by Coounty.